An equilateral triangle with sidelength $L$ can be tiled by trapezoids with sidelengths $2,1,1,1$.
What are the possible values for $L$?
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2 Answers
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An equilateral triangle with sidelength $L$ can be divided into $L^2$ equilateral triangles with sidelength $1$. A trapezoid with sidelengths $2,1,1,1$ covers three of these triangles, so $L^2$ must be divisible by $3$. $L$ has to be an integer (because if it is tiled by the trapezoids, its sidelength must be the sum of sidelengths of the trapezoids), so $L$ is divisible by $3$.
Note that three of these trapezoids can tile an equilateral triangle with sidelength $3$. Then, for any equilateral triangle with a sidelength that is a multiple of $3$, we can divide it into triangles with sidelength $3$ and tile each of those with the trapezoids.
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They can be arranged like so:
which means L is
any multiple of 3 (green is the base triangle to show further tiling)
